A Revival of Sylvester’s Wave Theory of Partitions

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Let p(n) denote the number of partitions of the integer n. The first exact formula for p(n) was given by Hardy and Ramanujan in 1918. Their formula is a divergent series, which can be truncated in the appropriate place to give the exact value of p(n). Rademacher improved Hardy and Ramanujan's formula in 1938 to a rapidly converging infinite series. In early 2011, Ono and Bruinier announced a new formula which expresses p(n) as a finite sum of algebraic numbers.

The Hardy-Ramanujan-Rademacher formula is really a statement about the coefficients of a certain modular form whose coefficents happen to be the values of p(n). The Ono-Bruinier formula expresses p(n) as a sum of singular moduli of a certain weak Maass form that can be described in terms of the Dedekind eta function and the quasimodular Eisensten series E_2. Although p(n) is clearly a combinatorial function, neither of these formulas is combinatorial.

In this talk, I will attempt to show that J. J. Sylvester (1857) and J. W. L. Glaisher (1909) were well on their way to finding a combinatorial formula for p(n), and that a revival of their work combined with the power of modern computers could lead to a new formula for p(n).


DIMACS - Rutgers Summer Seminar Series in Combinatorics and Experimental Math


Piscataway, NJ